This document discusses the definition, notation, history, and applications of derivatives. It begins by defining a derivative as the instantaneous rate of change of a quantity with respect to another. It then discusses differentiation, derivative notation, and the history of derivatives developed by Newton and Leibniz. Real-life applications described include using derivatives in automobiles, radar guns, and analyzing graphs. Derivatives are also applied in physics to calculate velocity and acceleration, and in mathematics to find extreme values and use the Mean Value Theorem.

2. PRESENTATION
Submitted to: MAM MARIAASHRAF
Department: BSCS
Section: “A”
GUJRAT INSTITUTE OF MANAGEMENT
SCIENCES
CALCULUS AND ANALYTICAL
GEOMETRY

3. APPLICATIONS OF
DERIVATIVES

4. CONTENTS
1. Definition of Derivative
2. Differentiation
3. Derivative notation
4. History
5. Real life Applications
6. Applications of derivatives in various field
7. Conclusion

5. Definition of derivative:
 The exact rate at which one quantity changes
with respect to another is called derivative.
 So Mathematically, derivative is a slope of
curve at a point on a curve.
 Derivative is also called Instantaneous rate of
change.
 Function of derivative represents infinitely
small change with respect to variable.

6. Differentiation:
 The process that is used to find a derivative is
called differentiation.
 The reverse process is known as
antidifferentiation
 So the elementary theorem of calculus states
that antidifferentiation is same as integration.
The derivative of f(x) by definition is defined
as:
f’(x) = lim f (x+h) – f(x) ( limit h 0)
h

7. Derivative notation:
Derivative Notation
Function Newton Form Leibniz Form
f(x) f ’(x) d (f(x))
dx
y y’(y prime) d (y)
dx
x + 1 (x + 1)’ d (x + 1)
dx

8. History
 Modern differentiation and derivatives are
ascribe to Issac Newton and Gottfried Leibniz.
 At that time they develop the elementary
theorem of calculus in 17th century. This relates
differentiation and integration in ways which
transfigure the procedure for computing areas
and volume.
 Newton work would not have been possible
without Isacc Barrows effort he began early
development of the derivative in 16th century.

9. Real life Applications of derivatives:
 Automobiles
 Radar guns
 graphs

10. Automobiles
 When we talk about an automobile there is
an odometer and a speedometer in it. So
these two gauges work in tandem and let the
driver to determine his speed and distance
that has been traveled. Electronic versions of
gauges use derivatives to transform data sent
to electronic motherboard from tires to miles
per hour (MPH) and distance (KM).

11. Radar Guns
 Keeping in mind about automobile theme, all
police officers who use radar guns are taking
benefit of the easy use of derivatives. When a
radar gun is pointed and fired at your care on
highway. the gun is capable to determine time
and distance at which radar was able to hit
certain section of vehicle. When we use
derivative it is able to calculate speed of car at
which it was going and also report distance of
the car was from the radar gun.

12. Graphs:
 The most common application of
derivative is to analyze data that can be
calculated by many different fields. So
when using derivative it is able to
calculate gradient at any point of graph.

13. Applications of Derivatives in Various
fields:
 Physics
 Mathematics
Derivatives in Physics:
• When we talk about physics, the derivative of
displacement of a moving body with respect to time is
velocity of body, and derivative of velocity with respect to
time is acceleration.
• Newtons second law states the derivative of momentum
of a body equal the force applied to body.

14. Derivatives in Mathematics:
The most common use of the derivatives in
Mathematics is to study functions as follow:
 Extreme values of a function
• Absolute maximum
• Absolute minimum
• Local maximum
• Local minimum
 The Mean Value theorem
 Monotonic functions

15. Extreme values:
 Absolute maximum:
function has absolute maximum value on domain
at point C if
f(x) <= f(c) for all x in D
 Absolute minimum:
function has absolute minimum value on
domain at point C if
f(x) >= f(c) for all x in D

16. Extreme values:
 Local maximum:
function has local maximum value at a interior
point C of its domain if
f(x) <= f(c) for all x in D
 Local minimum:
function has local minimum value at a interior
point C of its domain if
f(x) <= f(c) for all x in D

17.  Mean value theorem:
Suppose when y = f(x) is continue us on closed
interval [a,b] and at least one point c in (a,b) at
which
f(b) - f(a) =f’(c)
b - a
 Monotonic functions:
A function which is increasing or decreasing at a
Interval I is called monotonic on I.

18. Conclusion:
 Derivatives are repeatedly used in routine to
help to measure how much changes are
there. they are used by government in
population censuses, numerous types of
science. when we know how to use
derivative and how to apply them in
everyday life it can be a pivotal part of
profession so this means that learning early
is always a adequate thing.